Jean-Pierre Serre

Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology.

Foundational work in algebraic geometry, and the Weil conjectures

In the 1950s and 1960s, a fruitful collaboration between Serre and the two years younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were FAC (Faisceaux Algébriques Cohérents, on coherent cohomology) and GAGA.

Related Topics:
1950s - 1960s - Alexander Grothendieck - Weil conjectures - FAC - Coherent cohomology - GAGA

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Serre had early on perceived a need to construct more general and refined cohomology theories to tackle these conjectures. In simple terms, the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology, with integer coefficients. Amongst Serre's early candidate theories (1954/55) was one based on Witt vector coefficients.

Related Topics:
Cohomology - Coherent sheaf - Finite field - Singular cohomology - Witt vector

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Grothendieck in SGA4 eventually delivered a full technical development. Around 1958 Serre had suggested that isotrivial covers of algebraic varieties should be important — those that become trivial after pullback by a finite covering map. This was one important step towards the eventual étale covering theory.

Related Topics:
SGA4 - 1958 - Covering map - étale covering

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In the later developments Serre was sometimes a source instead of counterexamples to over-optimistic extrapolations. He also had a close working relationship with Pierre Deligne, who eventually finished the proof of the Weil conjectures.

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